Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in C_0^\infty(0,T)$. “Passing to the limit” we get $$\int_0^T u(t)\varphi'(t)\ dt=-\int_0^T v(t)\varphi(t)\ dt.\tag{3}$$ My question is: How to justify this passage to the limit? Remark: this passage to the limit is a step of the proof that “generalized derivatives are compatible […]

$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$ The results is obvious enough, but how to prove this

I’m just trying to prove that each open set in $\mathbb{R}^2$ is measurable using the product measure derived from the Lebesgue measure. This is the first thing that came to mind because I recall that open sets are countable unions of disjoint open intervals, but I was wondering if there is an elementary way to […]

I have this exercise: Suppose $f$ is a real valued continuous function on $[a,b]$. Show that $f(B)$ is a Borel set for every Borel subset $B$ of $[a,b]$. Hint: Consider the collection $M$ of all subsets $A$ of $[a,b]$ for which $f(A)$ is a Borel set. Show that $M$ is a sigma-algebra. I am struggling […]

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me that there is nothing special about the coordinate axes, so indeed if some direction $v\in \partial B(0,1)$ is fixed, we must have […]

I have reviewed Ayman Houreih’s proof for the limit of the $L_p$ norm as $ p \rightarrow 0$ at "Scaled $L^p$ norm" and geometric mean. While I have found the outline of the proof very helpful, I have a question regarding the applicability of LDCT: The proof mentions $\frac{|f|^q – 1}{q}$ as the dominating function. […]

Question from an exam sample I’m studying for: Suppose $\left(X,\mathcal{F},\mu\right)$ is a measure space and $f_{n}:\left(X,\mathcal{F}\right)\to\mathbb{R}$ a sequence of non-negative integrable functions such that $\int_{X}f_{n}d\mu=1$ . Is is it necessarily true that $\frac{1}{n}f_{n}$ converges almost everywhere to $0$ ? what about $\frac{1}{n^{2}}f_{n}$ ? My line of thought: From Fatou’s llema we got that: $$\int\limits _{X}\liminf_{n\to\infty}\frac{1}{n}f_{n}d\mu\geq\liminf_{n\to\infty}\int\limits […]

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any subsegment $[a,b]\subset[0,1]$ we’d have $\mu(A\cap [a,b])=\mu(B\cap [a,b])$? More particular case is that for any $x\in[0,1]$ we have $\mu(A\cap [0,x])=\mu(B\cap [0,x])$ and hence $\mu(A\cap [0,x])=\frac{x}{2}$. Metaphor. Such decomposition can be assosiated […]

A question from my homework: Suppose $\mu$ is a measure on the real line w.r.t to the Borel $\sigma$-algebra such that $\forall x \in \mathbb{R}$ $\mu(\{x\})=0 $. is $\mu$ necessarily absolutely continuous w.r.t. to the Lebesgue measure? We say $\mu$ is absolutely continuous w.r.t. to $m$ if for every measurable set $E$ such that $m(E)=0$ […]

Let $X$ be a Hausdorff space (or let’s even assume it is metrizable). A strictly positive measure on $X$ assigns positive measure to any non-empty open subset of $X$. What conditions on $X$ ensure the existence of a strictly positive probability (or equivalently $\sigma$-finite) measure? Is there a standard way to construct such a measure […]

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